[Paper Review] Integrable Gradient Flows and Morse Theory

Ivan Dynnikov, А. П. Веселов|arXiv (Cornell University)|Jun 9, 1995

Homotopy and Cohomology in Algebraic Topology13 references21 citations

TL;DR

This paper introduces integrable gradient flows for Morse functions on classical Riemannian manifolds, particularly Lie groups and symmetric spaces, showing that generic height functions on these spaces are perfect Morse functions with explicitly solvable gradient flows. The key contribution is a geometric cell decomposition of the manifold via these flows, leading to an elementary proof of Vassiljev's theorem on the flag join of Grassmannians and a realization of the homology of unitary groups via Schubert cycles.

ABSTRACT

Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the homology requires, and the corresponding gradient flow can be described explicitly. This gives an explicit cell decomposition and geometric realization of the homology for such a manifold. As another application of the integrable Morse functions we give an elementary proof of Vassiljev's theorem on the flag join of Grassmannians.

Motivation & Objective

• To establish conditions under which Morse functions on Riemannian manifolds admit integrable gradient flows.
• To demonstrate that generic height functions on classical Lie groups and symmetric spaces are perfect Morse functions.
• To provide an explicit geometric cell decomposition of such manifolds using integrable gradient flows.
• To give an elementary proof of Vassiljev's theorem on the flag join of Grassmannians using integrable Morse theory.
• To realize the homology of unitary groups via Schubert cycles constructed from invariant subspaces in the flag join.

Proposed method

• Define integrable Morse functions as Morse functions whose gradient flow equations are explicitly solvable.
• Use the trace function $ f = \frac{1}{3}\operatorname{Tr}(X^3) $ on the sphere of traceless symmetric matrices to construct an integrable gradient flow.
• Introduce barycentric coordinates $ a_i = \frac{\lambda_{i+1} - \lambda_1}{\lambda_n - \lambda_1} $ to transform the flow into a generalized Volterra chain.
• Solve the Volterra-type system via change of variables $ b_i = \sum_{k=1}^i a_k $, yielding $ \frac{db_i}{d\tau} = b_i(b_i - 1) $, which integrates explicitly.
• Construct cycles in $ U(n) $ from Schubert cells in Grassmannians $ G_k(\mathbb{C}^n) $ via unitary operators preserving a subspace and acting as $ -I $ on its complement.
• Use the explicit flow dynamics to show that the flag join of Grassmannians admits a smooth structure diffeomorphic to a standard sphere.

Experimental results

Research questions

• RQ1 Can height functions on classical Lie groups and symmetric spaces be shown to be perfect Morse functions with integrable gradient flows?
• RQ2 How can the gradient flow of $ f = \frac{1}{3}\operatorname{Tr}(X^3) $ on the sphere of symmetric matrices be explicitly integrated?
• RQ3 What is the geometric significance of the resulting cell decomposition in terms of Grassmannian embeddings?
• RQ4 Can the Vassiljev-Mahowald decomposition of $ H_*(U(n)) $ be derived from integrable gradient flows?
• RQ5 Does the flag join of Grassmannians admit a smooth structure compatible with the flow-induced decomposition?

Key findings

• The gradient flow of $ f = \frac{1}{3}\operatorname{Tr}(X^3) $ on the traceless symmetric matrix sphere is integrable and decomposes the sphere into simplexes with vertices in embedded Grassmannians.
• The flow in barycentric coordinates satisfies the generalized Volterra chain equation $ \frac{d}{d\tau}a_i = a_i\left(\sum_{k=1}^{i-1}a_k - \sum_{l=i+1}^{n-1}a_l\right) $, which is explicitly solvable.
• The solution $ b_i = \frac{1}{1 - c_i e^{\tau}} $ for $ b_i = \sum_{k=1}^i a_k $ confirms the integrability and provides a complete description of the flow.
• The flag join of Grassmannians $ \Theta_n(\mathbf{k}) $ admits a smooth structure diffeomorphic to the standard sphere, with all Grassmannian embeddings being smooth.
• The Vassiljev-Mahowald decomposition $ H_i(U(n)) \cong \bigoplus_{k=0}^n H_{i-k^2}(G_k(\mathbb{C}^n)) $ is geometrically realized via cycles constructed from Schubert cells.
• The construction provides an elementary proof of Vassiljev’s theorem using the dynamics of integrable gradient flows.